Rr*Rl f H*H, x*iy*(y1iy for /: H*H

Annales Academir Scientiarum ^Fennicrc Series A. I. Mathematica

Volumetr 8, 1983, 187-191

THE DILATATION OF BEURTING-AHLFORS EXTENSIONS OF QUASISYMMETRIC

FUNCTIONS

MATTI LEHTINEN

l.

lntroduction. The standard construction

for a

quasiconformal extension of a real quasisymmetric function

å

is the one introduced by Beurling and Ahlfors

[l].

They proved that a q-quasisymmettic

h

^has a g2-quasiconformal extension to the upper half-plane

H. A

careful examination of Beurling's and Ahlfors's estimation shows

that

pz can be replaced

by _Q't' for g

^close to one and

by

3pzl4

fot

large

a

I4l.

^{T. Reed [6] has given the bound 8g, which} is better than the previous ones

for large

g.

The bound

2q

^{has been announced} by wan-cai Lai _{[3], but} the cor- rectness of his proof ^{has been doubted} (cf. [7]). In ^this note we shall establish the dilatation bound 29, using elementary methods different from those of Lai.

It is well known that to every quasisymmetric

å

there exist one or more extremal extensions, i.e., quasiconformal maps

/: H*H ^with

^smallest maximal dilation among those quasiconformal self-maps

of

11 which agree

with ft

on the boundary.

In ^general, the explicit form of the extremal extensions of ^a given

å

^{is not known.}

If h is

q-quassymmetric

with q

close to one,

h

has a Beurling-Ahlfors ^extension with maximal dilatation ^{also close}

to

one. However,

K.

^Strebel _[9] ^{has shown by}

a simple normal family ^{argument that no} Beurling-Ahlfors ^{extension of} the g-quasi- symmetric

function h,h(x):x for x-O,h(x):qx, q>1, for x>0,

can ^have maximal dilatation arbitrarily ^close to the dilatation of the corresponding ^extremal (which is explicitly known

in

this case).

In this note, we shall complement strebel's result by showing that

if

q(å) is the the smallest number

q

for which å is g-quasisymmetric, then all Beurling-Ahlfors extensions

of h

have maximal dilatation at least q(å). As a consequence of this one sees that Beurling-Ahlfors extensions can have considerably larger dilatation than the corresponding extremals. Consider,

for

^{instance, the affine stretching}

x*iy*(y1iy

^{of a square with sides parallel to the} coordinate axes.

If

the map is lifted with the aid of two conformal transformations to ^a K-quasiconformal map

f

^:

^H*H,

^{the boundary}

function h--71nt

satisfies

g(h):l(():(l/16)

exp(zK)

-112+o(l)

^{[5, p. 81].}

2. The upper bound. Given an increasing g-quasisymmetric h:

Rr*Rl

^and

doi:10.5186/aasfm.1983.0817

188 Maru LBHnNrx

a

constant

r=0,

the Beurling-Ahlfors extension Jf,,,:

H-H

^is constructed as

follows:

for z:x*iyۊ,

^set

!x(z) -

0

v§ (z)

- t

^{h@* r) ctt,}

-.v and

2fo."Q)

:

a(z)*

fr(z)+ir(a(z)-

^p(z)).

Since linear transformations do not affect the property of being p-quasisymmetric

or the dilatation of a quasiconformal mapping, we may make certain simplifying assumptions when estimating the dilatation quotient

of fo,, ^at

^an

^arbitrary

^z.

First, we may suppose

that å

is normalized, i.e., satisfies

lu(0):g

and

å(l):1,

and secondly, we may restrict ourselves to the point

z:i.

^These conventions are obeyed in the rest of this section.

The dilatation quotient

D

of

fo,, ^at i

^satisfies

(1)

2r (( + d@ +D-1)

:

⁽¹ ₊ r')(((1 + 6z) + (-r(1 _{+ ry\)}

+zlt -

^(q)(r

-

^r'),

where

(:a*18*,(:urla*,4:§nl fr*.Sinceft

isnormalized,oneeasilygets

a*(i):1, 0,(i): -h(-l),a,(i):t- lihltldt

^and

fr,(i):h(-1)-/'-, h(t)dt. The

^q-quasi-

symmetry

of å

immediately yields A-a=€=A. ^{By a lemma of} Beurling and Ahlfors

ll,

^{p. 137),}

(2) h(t) dt

*

^p,

where

,i:(1+S)-1

and

p--qL. It

follows

that (

and

4

^{both lie in} the interval

U", pl.

Remark.

The bounds in (2) are not the best possible. Equality in, say, the right hand side of (2) holds for the non-quasisymmetric majorant

for

normalized g-quasiconformal functions introduced by R. Salem [8] and later studied by K. Gold- berg

[2].

By applying (2) repeatedly to properly chosen subintervals

of

[0,

l]

^one

obtains slightly sharper bounds. For our purposes the improvement is negligible, and will not be taken into account.

The range

of

(€,q,

Q

^{is further} restricted by the following

Lemma. If h

is a normalizetl g-quasisymmetric Junction, ^then

v

{

^{h@* t) ctt,}

0

,,*,/

(3) and (4)

{ D_{

i

^(t)

ctt=

- ^, _l

h(t) dt

=

^{(2s+ h(t) dt}

-2qh(-

¹⁾

h(t) dr.

Proaf. The q-quasisymmetry

of h

^implies

h (t)

- ^h(tr-

^L)12)

₌

^a(h((r

-

^L)12)

- ^h(-

^D)

The dilatation of Beurling-Ahlfors extensions of quasisymmetric 189

for

all t€l-1,11,

and (3) follows

by

integration. Also, since

ft is

normalized,

h(t)=--ah(-r)

^for

all

^r€[0,

^1],

^{and (4) follows.}

When we use the notation adopted above, (3) and (4) ^become (s)

and (6)

( =

^{((zq+ 1) q}

-

^1)/(1-

^o

( =

^q(l

-ril] -O.

when estimafing

D

from ^above, we simplify computations by taking

r:1.

This

is

motivated by computations

in

_[4], which indicate that

for large

Q, fo,,

with r

^close

^to

^{1 are the} best choices for small maximal dilatation.

Theorenr

1. ^The

maximal dilatation

of

the Beurling-Ahlfors ^extension

fnt

^{of a} q'quasisymmettic function

h

is at most ^2q'

Proof.

It

suffices

to

^prove

D=2Q,

where

D ^is

the dilatation quotient ^at

i ^of fo,,

^for a normalized q-quasisymmetric

ft'

^{By (1),}

D +

D-r :

(((1.u6z)+(-1(1

+q\)lG +ri :

F((, _{rt, O.}

Since we may,

if

need be, replace

h by g,g(x):h(-x)lh(-l),

^{we may assume}

(>1.

With this restriction,

.F

is maximized

for fixed ((,4)

^either

at c:l

^or

ut th" lu.g"rt ^{possible value}

of (.

The first alternative can take place only

for (=4'

In this case we have

Fr(€, 4,

\ :

^{@2 + 2(q}

-

⁽²

-2)

^{I G + d'z'}

The nominator is a quadratic polynomial

in _4,

and

it

is negative

for 4:0

^and

4:1.

ConsequentlY,

F(t,4, l) =

^{F((, 1,}

l) :

₄₊

_€-'

=

)"+ A-1

for ).<(<4,

and

D<).-r:Q*l=2a.

To estimate

F fot (

large, we divide the square _{(C, rDli= _C= ^p, )'=r1= trt\ into four domains

Tr,Tl,T, and Tr. 7, ^is

the triangle with vertices (p,)'),

(p,lt)

and (v, v), where

v:(q*1)/(3e+ l), Ti is 7,

reflected ovet

(:4,7,

^{is the triangle}

with vertices ^{(,1, v), (v,}

v)

and

(A,1),

and

7,

^is the trapezoid with vertices

(i'

l')' 0t,)"),

(v,v) and (/.,v). If (i,q)€Tr, D+D-|=F(E,4,q),

and we can ^compute

p Fr(t,

I,

Q)

:

(Z/f +Ztll

-

^(o'

^t'

^{+ q' + l))l G + r»'.}

Since Fr((,0, Q)=0 and

F,

^{can change sign at most once}

^fot

^{4>0, the maxima}

of F((,rt,S) io 7r

areattainedonthelines

Ir: (:4 or LziaC+(2a+l)4:q*l'

Since ,tr((,

t, d:((+€-')(q*g-\12

^the

^condition

^F(C,

^(,

^{s)€F(v, v,}

g)

holds

on 21n71. A

lengthy but elementary computation shows

that F(t,v'q)=2p*

(2s)-'

is true

for ^q=1. on L,

we estimate

Fr(q):F(l*n-t-(2*a-\4,4,

^Q)'

By computation,

(l

-

^{t»' k I 2 t ö F', (tD}

: _-

(2 s' + ² s + ^{l) qz}

*

^{(4 qz}

*

^a s + 2) tt

-

^{k2 + 2}

^d'

190 Mnrrr LrnrmrN

Since Fi(O)=0, F{(l)>Q,

F,

is maximized

on LraT,

either

at 4:y ^{or at q:).}

The former case has already been treated, and

a

direct computation shows that Fr(),):p1U,/.,

g)

is at most

2O*ed-, if

^g>1.

Since (=1,

F(€,4,

d=FO5 L d for ^(=ry.

^Hence

^the

estimation of

F(t,rl, q) in fi

^reduces to the work done

in

2,.

Next consider ((,

DeTz.

^{By (6),}

F(t, q, O

=

^{F(1, 4,}

^s0 -dl! -

^O)

:

Fr(€, q).

By computation,

(1

- c),{n,+z#r_L)

q(1 +(r)(1

+o

L-€

D +D_L

of

fo,,

^&t

i

satisfies

-

^{a (€, rilr + b (€,}

^rilr-t,

+

a(1

-4il'

⁾

and we see

that (Fr)r((,0)=0

and

(Fr),

can change sign at most once for fixed (.

It

follows that the maximum

of F,

is attained either

on TloT, or on 4:y. ^fhs

former possibility has already been considered. Reasoning exactly as above, we see

that

Fr((,

v)

is maximized either

at 4:y

^or

^at (:).

By direct computation one verifies

(7) _Fr(i,

_v)

= 2o*ed-,.

To complete the proof, (€,

D€T,

^{has stin to be} considered. By (5), we have in this case

F(t, _ry,

0 =

^{F((, ry,}

((2e+t)4-ty1- ₀₎ :

F,11,41.

The proof

that Fr((,q)=20+Qp)-t in z,

is carried out as above: by the con- sideration

of (fr)e

the possible maximal set is reducedto

1:7

^and

^Trn7r,

^and

on (- 1

one only has to take care of the points

n:).

^and

^q:y.

Remark. As g**,

(7)

is

asymptotically true as an equality.

The

upper

limit 2g

cannot be improved without further restriction of the range

of

((, tt, g.

3. The lower bound. In deriving a condition relating the lower bound of maximal dilatations

of f^,,

^{to the deviation from symmetry}

of å, it

is natural

to

^consider

the smallest

q

for which

å

is q-quasisymmetric.

we

denote such

a s ^by

s@).

Theorem

2. Let h: Rr*Rr

be quasisymmetric. Then the maximal dilatation of euery

f^,,

is at least q(h).

Proof. For every

a=a(h)

there are points

x,t(Rr

such that

h (x + ^t)

-

^{h (x)}

:

p (h (x)

-

^{h (x}

-

^t)).

There is no loss of generality in By (1), the dilatation quotient D

assuming

that h

is norm alized and

h(-

1)

-t:

g.

2q (C + ri a (€, _ry)

:

(e

-

^{1), + (e(}

*

^rt)r,

2q(€ _*ry)',b (€, q)

-

^{(q + 1), + (e( -ry)v}

where

The dilatation of Beurling-Ahlfors extension sof quasisymmetric

It

follows that

for fixed

((, ₄₎

(D+O''1' ₌ 4a((,q\b((,q): Fn(6ti'

since

h is

p(å)-quasisymmetric, (5) and (6) imply that the distance

of ((,4)

from

T1

can be made arbitrarily ^small

if q

is chosen close

to

q(h). Considering ^(Fa){, one sees as before that

Fn(t, q)

z

^Fn(€, O

: ((e- t)4+(s* l)n+(e'-

112((z-v(-z))l@qz)

=

^{Fn(P, P)}

holds

in ^Tr.

By direct computation ^{one verifies}

(8)

^{FnQt, p)} =

(q* 8-t)'

as a strict inequality

for

g> 1. continuity

of

Fn together with ⁽⁸⁾ yield

D+D-L>

s(h)+p(h)-r

^{for the proper choice}

of

^q.

References

[l]

^{BrunrrNc, A., and}

^L.

^{V. Anrrons: The boundary} corr,espondence under quasiconformal mappings. - Acta Math' 96, 1956, 125-142.

[2] ^G9LDBERG, K.: A new definition for quasisymmetric functions. - Michigan Math. J' 21,1974, 49-62.

t3l Ln1, WnN-Cnr: A theorem of Beurling and Ahlfors. - Acta Math. sinica 22, 1979, 178-184 (Chinese)'

[4] LunrrNrN, M.: A real-analytic quasiconformal extension of a quasisymmetric function' - Ann.

Acad. Sci. Fenn. ^Ser. A I Math. 3, 1977,207-213'

[5] Lilryo, o., and K.

I.

VnuNrN: Quasiconformal mappings in the plane' - springer-verlag, Berlin-Heidelberg-New York, 1973'

[6] Rnno, T : Quasiconformal mappings with given boundary values. - Duke Math. J. 33, 1966, 459-464.

UI ^{Rrrcrr, E.: Review 30020 in Math. Reviews 81m' 1981' 4947.}

[8i ^{S41nru, R.: On some} singular monotonic functions which ^are strictly increasing. - Trans. Amer' Math. Soc. 53, 1943, 427-439.

[9] Srnrser, K.: On the existence of extremal Teichmueller mappings. - ^J. Analyse Math. 30' 1976' 464-480.

University of Helsinki Department of Mathematics SF-00100 Helsinki ¹⁰ Finland

Received 25 November 1982

191